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# Approach to the consistency and consensus of Pythagorean fuzzy preference relations based on their partial orders in group decision making

• * Corresponding author: Ze Shui Xu and Zhen Ming Ma

The first author is supported by NSF of Shandong Province grant: ZR2017MG027

• Although intuitionistic fuzzy preference relations have become powerful techniques to express the decision makers' preference information over alternatives or criteria in group decision making, some limitations of them are pointed out in this paper, then they are overcame by developed the group decision making with Pythagorean fuzzy preference relations (PFPRs). Specially, we provide a partial order on the set of all the PFPRs, based on which, a deviation measure is defined. Then, we check and reach the acceptably multiplicative consistency and consensus of PFPRs associated with the partial order and mathematical programming. Concretely, acceptably multiplicative consistent PRPRs are defined by the deviation between a given PFPR and a multiplicative consistent PFPR constructed by a normal Pythagorean fuzzy priority vector. Then acceptable consensus of a collection of PFPRs is defined by the deviation of each PFPR and the aggregated result from symmetrical Pythagorean fuzzy aggregation operators. Based on which, a method which can simultaneously modify the unacceptable consistency and consensus of PFPRs in a stepwise way is provided. Particularly, we also prove that the collective PFPR obtained by aggregating several individual acceptably consistent PFPRs with various symmetric aggregation operators is still acceptably consistent. Then, a procedure is provided to solve group decision making with PRPRs and a numerical example is given to illustrate the effectiveness of our method.

Mathematics Subject Classification: Primary: 68T35, 90B50, 62A86, 03E72.

 Citation: • • Table 1.  The modified PFPRs ${P}^{(h)}$ and their corresponding multiplicatively consistent PFPRs $\widetilde{P}^{(h)}, h = 0, 1, 2$

 ${P}^{(0)}$ $(0.7071, 0.7071)$ $(0.469, 0.8216)$ $(0.7969, 0.3674)$ $(0.9301, 0.2646)$ $(0.8216, 0.469)$ $(0.7071, 0.7071)$ $(0.8689, 0.2121)$ $(0.9618, 0.1449)$ $(0.3674, 0.7969)$ $(0.2121, 0.8689)$ $(0.7071, 0.7071)$ $(0.7583, 0.3873)$ $(0.2646, 0.9301)$ $(0.1449, 0.9618)$ $(0.3873, 0.7583)$ $(0.7071, 0.7071)$ ${P}^{(1)}$ $(0.7071, 0.7071)$ $(0.469, 0.8216)$ $\underline{(0.8559, 0.3462)}$ $(0.9301, 0.2646)$ $(0.8216, 0.469)$ $(0.7071, 0.7071)$ $(0.8689, 0.2121)$ $(0.9618, 0.1449)$ $\underline{(0.3462, 0.8559)}$ $(0.2121, 0.8689)$ $(0.7071, 0.7071)$ $(0.7583, 0.3873)$ $(0.2646, 0.9301)$ $(0.1449, 0.9618)$ $(0.3873, 0.7583)$ $(0.7071, 0.7071)$ ${P}^{(2)}$ $(0.7071, 0.7071)$ $(0.469, 0.8216)$ $(0.8559, 0.3462)$ $(0.9301, 0.2646)$ $(0.8216, 0.469)$ $(0.7071, 0.7071)$ $\underline{(0.9178, 0.2121)}$ $(0.9618, 0.1449)$ $(0.3462, 0.8559)$ $\underline{(0.2121, 0.9178)}$ $(0.7071, 0.7071)$ $(0.7583, 0.3873)$ $(0.2646, 0.9301)$ $(0.1449, 0.9618)$ $(0.3873, 0.7583)$ $(0.7071, 0.7071)$ $\widetilde{P}^{(2)}$ $(0.7071, 0.7071)$ $(0.469, 0.8209)$ $(0.856, 0.3462)$ $(0.9451, 0.1952)$ $(0.8209, 0.469)$ $(0.7071, 0.7071)$ $(0.9179, 0.2121)$ $(0.9627, 0.1136)$ $(0.3462, 0.856)$ $(0.2121, 0.9179)$ $(0.7071, 0.7071)$ $(0.7583, 0.3873)$ $(0.1952, 0.9451)$ $(0.1136, 0.9627)$ $(0.3873, 0.7583)$ $(0.7071, 0.7071)$ $\widetilde{P}^{(1)}$ $(0.7071, 0.7071)$ $(0.4691, 0.8209)$ $(0.8561, 0.3462)$ $(0.9452, 0.1952)$ $(0.8209, 0.4691)$ $(0.7071, 0.7071)$ $(0.9179, 0.2121)$ $(0.9627, 0.1136)$ $(0.3462, 0.8561)$ $(0.2121, 0.9179)$ $(0.7071, 0.7071)$ $(0.7583, 0.3873)$ $(0.1952, 0.9452)$ $(0.1136, 0.9627)$ $(0.3873, 0.7583)$ $(0.7071, 0.7071)$ $\widetilde{P}^{(0)}$ $(0.7071, 0.7071)$ $(0.47, 0.8206)$ $(0.8587, 0.3443)$ $(0.9452, 0.1922)$ $(0.8206, 0.47)$ $(0.7071, 0.7071)$ $(0.9192, 0.2111)$ $(0.9628, 0.1122)$ $(0.3443, 0.8587)$ $(0.2111, 0.9192)$ $(0.7071, 0.7071)$ $(0.7593, 0.3851)$ $(0.1922, 0.9452)$ $(0.1122, 0.9628)$ $(0.3851, 0.7593)$ $(0.7071, 0.7071)$

Table 2.  Individual preference information from three decision makers

 $P^{(1)}$ $(0.7071, 0.7071)$ $(0.9487, 0.1)$ $(0.8367, 0.3162)$ $(0.1, 0.9487)$ $(0.7071, 0.7071)$ $(0.7746, 0.4472)$ $(0.3162, 0.8367)$ $(0.4472, 0.7746)$ $(0.7071, 0.7071)$ $P^{(2)}$ $(0.7071, 0.7071)$ $(0.7746, 0.3162)$ $(0.8367, 0.4472)$ $(0.3162, 0.7746)$ $(0.7071, 0.7071)$ $(0.7071, 0.3162)$ $(0.4472, 0.8367)$ $(0.3162, 0.7071)$ $(0.7071, 0.7071)$ $P^{(3)}$ $(0.7071, 0.7071)$ $(0.7746, 0.4472)$ $(0.8944, 0.3162)$ $(0.4472, 0.7746)$ $(0.7071, 0.7071)$ $(0.7746, 0.3162)$ $(0.3162, 0.8944)$ $(0.3162, 0.7746)$ $(0.7071, 0.7071)$

Table 3.  Iterative process of the proposed method for the acceptable concensus

 $h$ $(D(P^{(1h)}, P_{Agg}^{(h)}), D(P^{(2h)}, P_{Agg}^{(h)}), D(P^{(3h)}, P_{Agg}^{(h)}))$ $l_0$ $i_0, j_0$ 0 $(0.3589, 0.2902, 0.2578)$ 1 $(1, 2)$ 1 $(0.271, 0.2774, 0.2244)$ 1 $(2, 3)$ 2 $(0.2205, 0.2606, 0.2291)$ 2 $(1, 3)$ 3 $(0.2175, 0.2288, 0.2203)$ 2 $(2, 3)$ 4 $(0.2037, 0.1762, 0.1986)$ 2 $(1, 2)$ 5 $(0.1943, 0.1377, 0.1875)$ 1 $(1, 3)$ 6 $(0.1493, 0.1540, 0.1598)$ 3 $(1, 2)$ 7 $(0.1188, 0.1355, 0.1076)$ 2 $(1, 3)$ 8 $(0.1017, 0.0787, 0.0814)$ 1 $(1, 2)$ 9 $(0.0780, 0.0628, 0.0757)$

Table 4.  Individual preference information from three decision makers

 $P^{(1)}$ $(0.707, 0.707)$ $(0.707, 0.447)$ $(0.837, 0.316)$ $(0.707, 0.548)$ $(0.447, 0.707)$ $(0.707, 0.707)$ $(0.775, 0.447)$ $(0.548, 0.775)$ $(0.316, 0.837)$ $(0.447, 0.775)$ $(0.707, 0.707)$ $(0.548, 0.775)$ $(0.548, 0.707)$ $(0.775, 0.548)$ $(0.775, 0.548)$ $(0.707, 0.707)$ $P^{(2)}$ $(0.707, 0.707)$ $(0.775, 0.316)$ $(0.894, 0.447)$ $(0.775, 0.548)$ $(0.316, 0.775)$ $(0.707, 0.707)$ $(0.707, 0.316)$ $(0.548, 0.837)$ $(0.447, 0.894)$ $(0.316, 0.707)$ $(0.707, 0.707)$ $(0.632, 0.775)$ $(0.548, 0.775)$ $(0.837, 0.548)$ $(0.775, 0.632)$ $(0.707, 0.707)$ $P^{(3)}$ $(0.707, 0.707)$ $(0.775, 0.447)$ $(0.894, 0.316)$ $(0.837, 0.447)$ $(0.447, 0.775)$ $(0.707, 0.707)$ $(0.775, 0.316)$ $(0.447, 0.837)$ $(0.316, 0.894)$ $(0.316, 0.775)$ $(0.707, 0.707)$ $(0.447, 0.548)$ $(0.447, 0.837)$ $(0.837, 0.447)$ $(0.548, 0.447)$ $(0.707, 0.707)$

Table 5.  Modified individual preference information from three decision makers

 $P^{(1, 44, 0)}$ $(0.707, 0.707)$ $(0.767, 0.341)$ $(0.89, 0.149)$ $(0.88, 0.136)$ $(0.341, 0.767)$ $(0.707, 0.707)$ $(0.769, 0.316)$ $(0.748, 0.271)$ $(0.149, 0.89)$ $(0.316, 0.769)$ $(0.707, 0.707)$ $(0.626, 0.549)$ $(0.136, 0.88)$ $(0.271, 0.748)$ $(0.549, 0.626)$ $(0.707, 0.707)$ $P^{(2, 44, 0)}$ $(0.707, 0.707)$ $(0.775, 0.316)$ $(0.894, 0.154)$ $(0.883, 0.128)$ $(0.316, 0.775)$ $(0.707, 0.707)$ $(0.772, 0.322)$ $(0.76, 0.271)$ $(0.154, 0.894)$ $(0.322, 0.772)$ $(0.707, 0.707)$ $(0.624, 0.549)$ $(0.128, 0.883)$ $(0.271, 0.76)$ $(0.549, 0.624)$ $(0.707, 0.707)$ $P^{(3, 44, 0)}$ $(0.707, 0.707)$ $(0.764, 0.327)$ $(0.894, 0.155)$ $(0.874, 0.13)$ $(0.327, 0.764)$ $(0.707, 0.707)$ $(0.775, 0.316)$ $(0.756, 0.274)$ $(0.155, 0.894)$ $(0.316, 0.775)$ $(0.707, 0.707)$ $(0.628, 0.548)$ $(0.13, 0.874)$ $(0.274, 0.756)$ $(0.548, 0.628)$ $(0.707, 0.707)$ $\widetilde{N}^{(44)}$ $(0.707, 0.707)$ $(0.75, 0.316)$ $(0.894, 0.149)$ $(0.883, 0.128)$ $(0.316, 0.775)$ $(0.707, 0.707)$ $(0.775, 0.316)$ $(0.76, 0.271)$ $(0.149, 0.894)$ $(0.316, 0.775)$ $(0.707, 0.707)$ $(0.628, 0.548)$ $(0.128, 0.883)$ $(0.271, 0.76)$ $(0.548, 0.628)$ $(0.707, 0.707)$ $P^{(44, 0)}$ $(0.707, 0.707)$ $(0.769, 0.328)$ $(0.893, 0.153)$ $(0.879, 0.131)$ $(0.328, 0.769)$ $(0.707, 0.707)$ $(0.772, 0.318)$ $(0.755, 0.272)$ $(0.153, 0.893)$ $(0.318, 0.772)$ $(0.707, 0.707)$ $(0.626, 0.549)$ $(0.131, 0.879)$ $(0.272, 0.755)$ $(0.549, 0.626)$ $(0.707, 0.707)$

Table 6.  The modified individual preference information from three decision makers by Xu's method 

 $\widetilde{P}^{(1)}$ $(0.707, 0.707)$ $(0.71, 0.445)$ $(0.01, 0.056)$ $(0.011, 0.553)$ $(0.445, 0.71)$ $(0.707, 0.707)$ $(0.047, 0.013)$ $(0.512, 0.802)$ $(0.056, 0.01)$ $(0.013, 0.047)$ $(0.707, 0.707)$ $(0.529, 0.619)$ $(0.553, 0.011)$ $(0.802, 0.512)$ $(0.619, 0.529)$ $(0.707, 0.707)$ $\widetilde{P}^{(2)}$ $(0.707, 0.707)$ $(0.71, 0.445)$ $(0.01, 0.056)$ $(0.011, 0.553)$ $(0.445, 0.71)$ $(0.707, 0.707)$ $(0.047, 0.014)$ $(0.512, 0.802)$ $(0.056, 0.01)$ $(0.014, 0.047)$ $(0.707, 0.707)$ $(0.529, 0.619)$ $(0.553, 0.011)$ $(0.802, 0.512)$ $(0.619, 0.529)$ $(0.707, 0.707)$ $\widetilde{P}^{(3)}$ $(0.707, 0.707)$ $(0.728, 0.447)$ $(0.011, 0.132)$ $(0.011, 0.553)$ $(0.447, 0.728)$ $(0.707, 0.707)$ $(0.046, 0.03)$ $(0.509, 0.804)$ $(0.132, 0.011)$ $(0.03, 0.046)$ $(0.707, 0.707)$ $(0.5, 0.573)$ $(0.553, 0.011)$ $(0.804, 0.509)$ $(0.573, 0.5)$ $(0.707, 0.707)$ $\widetilde{P}$ $(0.707, 0.707)$ $(0.716, 0.446)$ $(0.01, 0.074)$ $(0.011, 0.553)$ $(0.446, 0.716)$ $(0.707, 0.707)$ $(0.047, 0.018)$ $(0.511, 0.803)$ $(0.074, 0.01)$ $(0.018, 0.047)$ $(0.707, 0.707)$ $(0.519, 0.603)$ $(0.553, 0.011)$ $(0.803, 0.511)$ $(0.603, 0.519)$ $(0.707, 0.707)$
• Tables(6)

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